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Q1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) 4x^{2}–3x+7
Ans:
The equation 4x^{2}–3x+7 can be written as 4x^{2}–3x^{1}+7x0
Since x is the only variable in the given equation and the powers of x (i.e., 2, 1 and 0) are whole numbers, we can say that the expression 4x2–3x+7 is a polynomial in one variable.
(ii) y^{2}+√2
Ans:
The equation y^{2}+√2 can be written as y^{2}+√2y^{0}
Since y is the only variable in the given equation and the powers of y (i.e., 2 and 0) are whole numbers, we can say that the expression y2+√2 is a polynomial in one variable.
(iii) 3√t+t√2
Ans:
The equation 3√t+t√2 can be written as 3t^{1/2}+√2t
Though t is the only variable in the given equation, the powers of t (i.e.,1/2) is not a whole number. Hence, we can say that the expression 3√t+t√2 is not a polynomial in one variable.
(iv) y+2/y
Ans:
The equation y+2/y can be written as y+2y^{1}
Though y is the only variable in the given equation, the powers of y (i.e.,1) is not a whole number. Hence, we can say that the expression y+2/y is not a polynomial in one variable.
(v) x^{10}+y^{3}+t^{50}
Ans:
Here, in the equation x^{10}+y^{3}+t^{50}
Though the powers, 10, 3, 50, are whole numbers, there are 3 variables used in the expression
x^{10}+y^{3}+t^{50}.
Hence, it is not a polynomial in one variable.
Q2. Write the coefficients of x^{2} in each of the following:
(i) 2+x^{2}+x
Ans:
The equation 2+x^{2}+x can be written as 2+(1)x^{2}+x
We know that, coefficient is the number which multiplies the variable.
Here, the number that multiplies the variable x^{2} is 1
the coefficients of x^{2} in 2+x^{2}+x is 1.
(ii) 2–x^{2}+x^{3}
Ans:
The equation 2–x^{2}+x^{3} can be written as 2+(–1)x^{2}+x^{3}
We know that, coefficient is the number (along with its sign, i.e., – or +) which multiplies the variable.
Here, the number that multiplies the variable x^{2} is 1 the coefficients of x^{2} in 2–x^{2}+x^{3} is 1.
(iii) (π/2)x^{2}+x
Ans:
The equation (π/2)x^{2} +x can be written as (π/2)x^{2} + x
We know that, coefficient is the number (along with its sign, i.e., – or +) which multiplies the variable.
Here, the number that multiplies the variable x^{2} is π/2.
the coefficients of x^{2} in (π/2)x^{2} +x is π/2.
(iv)√2x1
Ans:
The equation √2x1 can be written as 0x^{2}+√2x1 [Since 0x^{2} is 0]
We know that, coefficient is the number (along with its sign, i.e., – or +) which multiplies the variable.
Here, the number that multiplies the variable x^{2} is 0, the coefficients of x^{2} in √2x1 is 0.
Q3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.
Ans:
Binomial of degree 35: A polynomial having two terms and the highest degree 35 is called a binomial of degree 35
Example: 3x^{35}+5
Monomial of degree 100: A polynomial having one term and the highest degree 100 is called a monomial of degree 100
Example: 4x^{100}
Degree of a polynomial
Q4. Write the degree of each of the following polynomials:
(i) 5x^{3}+4x^{2}+7x
Ans:
The highest power of the variable in a polynomial is the degree of the polynomial.
Here, 5x^{3}+4x^{2}+7x = 5x^{3}+4x^{2}+7x1
The powers of the variable x are: 3, 2, 1
The degree of 5x^{3}+4x^{2}+7x is 3 as 3 is the highest power of x in the equation.
(ii) 4–y^{2}
Ans:
The highest power of the variable in a polynomial is the degree of the polynomial.
Here, in 4–y^{2},
The power of the variable y is 2
The degree of 4–y^{2} is 2 as 2 is the highest power of y in the equation.
(iii) 5t–√7
Ans:
The highest power of the variable in a polynomial is the degree of the polynomial.
Here, in 5t–√7,
The power of the variable t is: 1
The degree of 5t–√7 is 1 as 1 is the highest power of y in the equation.
(iv) 3
Ans:
The highest power of the variable in a polynomial is the degree of the polynomial.
Here, 3 = 3 × 1 = 3 × x^{0}
The power of the variable here is: 0
The degree of 3 is 0.
Q5. Classify the following as linear, quadratic and cubic polynomials:
Ans:
We know that,
Linear polynomial: A polynomial of degree one is called a linear polynomial.
Quadratic polynomial: A polynomial of degree two is called a quadratic polynomial.
Cubic polynomial: A polynomial of degree three is called a cubic polynomial.
(i) x^{2}+x
Ans:
The highest power of x^{2}+x is 2
The degree is 2
Hence, x^{2}+x is a quadratic polynomial
(ii) x–x^{3}
Ans:
The highest power of x–x^{3} is 3
The degree is 3
Hence, x–x^{3} is a cubic polynomial
(iii) y+y^{2}+4
Ans:
The highest power of y+y^{2}+4 is 2
The degree is 2
Hence, y+y^{2}+4is a quadratic polynomial
(iv) 1+x
Ans:
The highest power of 1+x is 1
The degree is 1
Hence, 1+x is a linear polynomial.
(v) 3t
Solution:
The highest power of 3t is 1
The degree is 1
Hence, 3t is a linear polynomial.
(vi) r^{2}
Ans:
The highest power of r^{2} is 2
The degree is 2
Hence, r^{2 }is a quadratic polynomial.
(vii) 7x^{3}
Ans:
The highest power of 7x^{3} is 3
The degree is 3
Hence, 7x^{3} is a cubic polynomial.
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